UNIVERSITI PUTRA MALAYSIA

SAEID MOLLADAVOUDI

IPM 2013 6

SYMPLECTIC TECHNIQUES IN GEOMETRIC QUANTUM MECHANICS AND NONLINEAR QUANTUM MECHANICS

© COPYRIG

HT UPMSYMPLECTIC TECHNIQUES IN GEOMETRIC

QUANTUM MECHANICS AND NONLINEARQUANTUM MECHANICS

By

SAEID MOLLADAVOUDI

Thesis Submitted to the School of Graduate Studies, Universiti PutraMalaysia, in Fulfilment of the Requirements for the Degree of Doctor of

Philosophy

June 2013

© COPYRIG

HT UPM

COPYRIGHT

All material contained within the thesis, including without limitation text, logos,icons, photographs and all other artwork, is copyright material of Universiti PutraMalaysia unless otherwise stated. Use may be made of any material contained withinthe thesis for non-commercial purposes from the copyright holder. Commercial use ofmaterial may only be made with the express, prior, written permission of UniversitiPutra Malaysia.

Copyright c©Universiti Putra Malaysia

i

© COPYRIG

HT UPM

DEDICATIONS

To Somayeh

© COPYRIG

HT UPM

Abstract of thesis presented to the Senate of Universiti Putra Malaysia infulfilment of the requirement for the degree of Doctor of Philosophy

SYMPLECTIC TECHNIQUES IN GEOMETRIC QUANTUMMECHANICS AND NONLINEAR QUANTUM MECHANICS

By

SAEID MOLLADAVOUDI

June 2013

Chair: Associate Professor Hishamuddin B Zainuddin, PhD

Faculty: Institute for Mathematical Research

In this thesis we study the roles played by symplectic geometry in quantum mechan-

ics, in particular quantum dynamics and quantum information theory treated as two

separate parts. The common ground for both parts is the geometrical formulation

of quantum mechanics. In Chapter 2, we review the associated complex projective

Hilbert space of quantum pure states, with symplectic and Riemannian structures

and their roles in quantum dynamics and kinematics.

In Chapter 3, we motivate the idea of information-theoretic constraint on the dif-

ferentiable manifold of probability distributions through the maximum uncertainty

principle and the linear Schrodinger equation by introduction of the wavefunction.

In Chapter 4, we review both regular and singular symplectic reduction of a sym-

plectic manifold, which is acted upon properly and symplectically by a compact Lie

group.

Chapter 5 contains the author’s original contributions to the first part of the thesis.

In this chapter, by using the same information-theoretic discussion of the Chapter

iii

© COPYRIG

HT UPM

3 we propose a non-relativistic, spin-less, non-linear quantum dynamical equation,

with the Fisher information metric replaced by the Jensen-Shannon distance infor-

mation. Furthermore, we show that the non-linear Schrodinger equation is in fact

a Hamiltonian dynamics, namely it preserves the symplectic structure of the com-

plex Hilbert space. The projected dynamics on the corresponding projective Hilbert

space is derived and its properties are highlighted in further details.

Chapter 6 contains the author’s primary contributions to the second part of the

thesis. In particular, by using the singular symplectic reduction of the Chapter 4 we

explicitly construct the space of entanglement types of three-qubit pure states with

a specific (shifted) spectra of single-particle reduced density matrices. Moreover, we

obtain the image of the symplectic quotient under the induced Hilbert map, by using

local unitary invariant polynomials. Then the symplectic structure on the principal

stratum of the symplectic quotient is derived. Finally, it is discussed that other

lower dimensional strata are relative equilibria on the original manifold and their

stability properties are investigated under compact subgroups of the local unitary

transformations.

iv

© COPYRIG

HT UPM

Abstrak tesis yang dikemukakan kepada Senat Universiti Putra Malaysia sebagaimemenuhi keperluan untuk ijazah Doktor Falsafah

TEKNIK SIMPLEKTIK DALAM MEKANIK KUANTUMBERGEOMETRI DAN MEKANIK KUANTUM TAK LINEAR

Oleh

SAEID MOLLADAVOUDI

Jun 2013

Pengerusi: Profesor Madya Hishamuddin B Zainuddin, PhD

Fakulti: Institut Penyelidikan Matematik

Dalam tesis ini, kita kaji peranan yang dimainkan oleh geometri simplektik dalam

mekanik kuantum, khususnya dalam dinamik kuantum dan teori maklumat kuantum

yang dikaji sebagai dua bahagian berasingan. Yang menjadi asas sama-sama bagi

kedua-dua bahagian adalah formulasi bergeometri mekanik kuantum. Dalam Bab 2,

kita tinjau kembali ruang Hilbert unjuran kompleks bagi keadaan tulen kuantum,

bersama dengan struktur-struktur simplektik dan Riemannian dan peranan mereka

dalam dinamik dan kinematik kuantum.

Dalam Bab 3, kita bangunkan idea kekangan berteori maklumat ke atas manifold

terbezakan bagi taburan kebarangkalian melalui prinsip ketakpastian maksimum dan

persamaan linear Schrodinger dengan memperkenalkan fungsi gelombang. Dalam

Bab 4, kita tinjau semula penurunan simplektik nalar dan singular bagi manifold

simplektik yang ditindak secara wajar dan simplektik oleh satu kumpulan Lie yang

padat.

Bab 5 mengandungi sumbangan asli penulis bagi bahagian pertama tesis. Dalam

bab ini, menggunakan perbincangan berteori maklumat Bab 3, kita usulkan satu per-

v

© COPYRIG

HT UPM

samaan dinamik kuantum yang bukan kereatifan, tanpa spin dan tak linear, dengan

menggantikan metrik maklumat Fisher dengan maklumat jarak Jensen-Shannon.

Tambahan itu, kita tunjukkan bahawa persamaan Schrodinger tak linear adalah

sebenarnya dinamik Hamiltonan, iaitu ia mengekalkan struktur simplektik ruang

Hilbert kompleks. Dinamik yang terunjur ke atas ruang Hilbert terunjur yang

berpadanan diterbitkan dan sifatnya disorot dengan terperinci.

Bab 6 mengandungi sumbangan utama penulis ke bahagian kedua tesis. Khususnya,

dengan menggunakan penurunan simplektik singular dari Bab 4, kita bangunkan

secara eksplisit ruang jenis keterbelitan bagi keadaan tulen tiga-qubit dengan spek-

trum (teranjak) khas bagi matriks ketumpatan terturun zarah tunggal. Tambahan

lagi, kita memperoleh imej hasil bahagi simplektik di bawah pemetaan Hilbert ter-

aruh dengan menggunakan polinomial tak varian unitary setempat. Kemudian itu,

struktur simplektik di atas stratum utama hasil bahagi simplektik diterbitkan. Akhir

sekali, dibincangkan strata berdimensi lebih rendah adalah equilibria relatif di atas

manifold asal dan ciri kestabilan mereka dikaji di bawah subkumpulan padat bagi

transformasi unitary setempat.

vi

© COPYRIG

HT UPM

ACKNOWLEDGEMENTS

First and foremost, I would like to express my sincere gratitude to my thesis supervi-

sor, Prof. Hishamuddin Zainuddin, for all his support and patience during the years

of my PhD studies and for trusting me to pretty much direct my own research. His

insight in quantum mechanics and his passion for mathematical physics have always

been inspiring.

I would like to thank my friend and coauthor Chan Kar Tim for helpful discussions.

Also, partial financial support from the Malaysian Ministry of Higher Education

(MOHE), Fundamental Reseach Grant Scheme (FRGS), under the project no. 01-

04-10-887FR (vote no. 5523927), is thankfully acknowledged.

I also would like to thank my Committee Members, Prof. Twareque Ali, Prof. Adem

Kilicman and Prof. Isamiddin Rakhimov for their constructive remarks on previous

version of this thesis.

My very special thanks would certainly go to my loving and devoted parents, Maryam

and Yahya, and my mother-in-law, Kabootar, for their unconditional love, encour-

agement and support throughout my life. I would like to extend my gratitude to

my family members, especially my sister, Sara, my brother-in-law, Nima, and my

sister-in-laws, Zeinab and Soheila, and their families for all their support. I am

deeply indebted to my Grandma, Maman Zahra, who left us during my PhD stud-

ies. She taught me so many wonderful things and filled my childhood with beautiful

moments; may her soul rest in peace.

Last but not least, I am most grateful to my wife, Somayeh, for her companionship

on my journey. Our faithful love is the primary source of strength, inspiration and

motivation. This thesis would have not been possible without her presence and I

dedicate it to her from the bottom of my heart.

vii

© COPYRIG

HT UPM

I certify that a Thesis Examination Committee has met on 10 June 2013 to conduct the finalexamination of Saeid Molladavoudi on his thesis entitled “Symplectic Techniques in GeometricQuantum Mechanics and Nonlinear Quantum Mechanics” in accordance with the Universities andUniversity Colleges Act 1971 and the Constitution of the Universiti Putra Malaysia [P.U.(A) 106]15 March 1998. The Committee recommends that the student be awarded the Doctor of Philosophy.

Members of the Thesis Examination Committee were as follows:

Mohamad Rushdan Md. Said, PhDAssociate ProfessorInstitute for Mathematical ResearchUniversiti Putra Malaysia(Chairperson)

Adem Kilicman, PhDProfessorFaculty of ScienceUniversiti Putra Malaysia(Internal Examiner)

Isamiddin Rakhimov, PhDProfessorFaculty of ScienceUniversiti Putra Malaysia(Internal Examiner)

Syed Twareque Ali, PhD

Professor

Department of Mathematics and Statistics

Concordia University

Canada

(External Examiner)

NORITAH OMAR, PhD

Assoc. Professor and Deputy Dean

School of Graduate Studies

Universiti Putra Malaysia

Date: 2 August 2013

viii

© COPYRIG

HT UPM

This thesis was submitted to the Senate of Universiti Putra Malaysia and has beenaccepted as fulfilment of the requirement for the degree of Doctor of Philosophy.The members of the Supervisory Committee were as follows:

Hishamuddin B Zainuddin, PhDAssociate ProfessorInstitute for Mathematical ResearchUniversiti Putra Malaysia(Chairperson)

Norihan Bt Md Arifin, PhDAssociate ProfessorInstitute for Mathematical ResearchUniversiti Putra Malaysia(Member)

Zuriati Bt Ahmad Zukarnain, PhDAssociate ProfessorFaculty of Computer Science and Information TechnologyUniversiti Putra Malaysia(Member)

BUJANG BIN KIM HUAT, PhDProfessor and DeanSchool of Graduate StudiesUniversiti Putra Malaysia

Date:

ix

© COPYRIG

HT UPM

DECLARATION

I declare that the thesis is my original work except for quotations and citations whichhave been duly acknowledged. I also declare that it has not been previously, and isnot concurrently, submitted for any other degree at Universiti Putra Malaysia or atany other institution.

SAEID MOLLADAVOUDI

Date:

x

© COPYRIG

HT UPM

TABLE OF CONTENTS

Page

DEDICATIONS ii

ABSTRACT iii

ABSTRAK v

ACKNOWLEDGEMENTS vii

APPROVAL viii

DECLARATION x

LIST OF FIGURES xiii

LIST OF SYMBOLS AND ABBREVIATIONS xiv

CHAPTER

1 INTRODUCTION 1

1.1 Problem Statements 21.2 Objectives and Motivations 31.3 Hamiltonian Dynamics in Quantum Mechanics 7

2 GEOMETRIC QUANTUM MECHANICS 10

2.1 The Hilbert Space as Kahler Manifold 102.2 The Quantum Phase Space 162.3 Symplectic Structure and Quantum Dynamics 19

2.3.1 Quantum Evolution as Hamiltonian Dynamics 192.3.2 Quantum Dynamics as Classical Integrable System 23

2.4 Riemannian Structure and Quantum Kinematics 252.4.1 Uncertainty Principle 252.4.2 Metric of Quantum Phase Space 292.4.3 Measurement and Quantum Observables 32

2.5 Moment Map Formulation 34

3 INFORMATION-THEORETIC CONSTRAINT AND QUANTUMDYNAMICS 383.1 Information Geometry 383.2 Statistical Inference Method 403.3 Linear Quantum Dynamics 423.4 Kullback-Leibler Information Measure 44

3.4.1 Beyond the Linear Theory 463.5 Jensen-Shannon Information Measure 47

xi

© COPYRIG

HT UPM

4 SYMPLECTIC REDUCTION THEORY 51

4.1 Regular Symplectic Reduction 514.2 Singular Symplectic Reduction 60

4.2.1 Embedding of Symplectic Reduced Space 744.3 Relative Equilibria 77

5 GENERALIZED QUANTUM DYNAMICS 80

5.1 Generalized Quantum Dynamics 805.1.1 The Non-linear Schrodinger Equation 805.1.2 Many Quantum Particles in Higher Dimensions 835.1.3 Parametric Non-linear Schrodinger Equation 86

5.2 Dynamical Properties 875.2.1 Background 875.2.2 Hamiltonian Dynamics 885.2.3 Metric Structure 91

6 SYMPLECTIC QUOTIENT OF TRIPARTITE PURE STATES 93

6.1 Review of Local Unitary Action 936.2 Singular Reduction of Local Unitary Action 976.3 Principal Stratum of Symplectic Quotient 1036.4 Relative Equilibria for Three-Qubit Pure States 105

7 CONCLUSION AND OUTLOOK 110

7.1 Conclusions 1107.2 Future Works 115

REFERENCES 117

APPENDICES 124

BIODATA OF STUDENT 157

LIST OF PUBLICATIONS 158

xii